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\title{Study on the effection of moving mesh for the elliptic interface problems}

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\section{Introduction}
The elliptic interface problems is a widely interested in the fields of computational fluid dynamics, material science. ...  The immersed interface finite element is popular strategy when solving such kind of problems to resolve the interface sharply, however, it is complicated for constructing the proper finite elements. 
Recently, Li suggest a heuristic way to put the real grid point just on the interface, which avoid the using of the immersed finite elements. To archive high accuracy and high order convergence,
it is necessary to enforce the interface jump conditions
when solving the elliptic interface problems.


An well tuned adaptively deformed mesh can catch the interface
sharply, so that the elliptic equations with discontinuous coefficients can be solved efficiently without constructing the immersed finite elements. The presented strategy is actually a improved version of the traditional adaptive moving mesh method. Matched interface boundary(MIB) method wai initiallized by Zhao and Wei when solving the Holmholtz equation[47] and Maxwell's equation with material interfaces [48].

In this research, we are intendent to study the mesh shape effection under such circumstans. The structured cartesian mesh and unstructured meshes are compared for such kind of strategy. Another interested work
here is the comparesion of different mesh adaptation strategies for the moving mesh generation.


\section{The moving mesh method for model problems}
\subsection{The model problems}


\subsection{The moving mesh equaiton}

\subsubsection{The adaptive mesh deformation}

\subsubsection{The moving mesh}


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